Applied Mathematics: an Introduction to Scientific Computing

Luca Heltai (luca.heltai@sissa.it)

Gianluigi Rozza (gianluigi.rozza@sissa.it)

Stefano Piani (stefano.piani@sissa.it)

Giovanni Stabile (giovanni.stabile@sissa.it)
Course information
This is a Joint course, between SISSA PhD in Mathematical Analysis, Modeling, and Applications, Laurea Magistrale in Matematica, Laurea Magistrale in Data Science and Scientific Computing, and Master in High Performance Computing.
All lectures will take place live in room 128129, will be streamed online using the Zoom platform, and will be recorded live on YouTube.
The first lecture will be on the 5th of October 2021 at 16.30 in Room A128/A129 and on Zoom.
The following Zoom link will be used for all lectures:
https://sissait.zoom.us/j/85796873697?pwd=cm5HNGRidndvU05UTVAvRkdOdnNrdz09
Meeting ID: 857 9687 3697 Passcode: NumAna
Recordings for lectures of 20212022
Recordings for lectures of 20202021
All course material is available at
https://github.com/lucaheltai/numericalanalysis20212022 (this github repository)
If you are following the course, please (FILL THIS FORM)[https://forms.gle/8DyESWnfCmXMei3h8].
Syllabus 20212022
Four Modules of 12h each (1.5 CFU for each module), for a total of 48h, 6 CFU
Frontal Lectures
Module 1 (Basis of Numerical Analysis  Part I  Prof. Luca Heltai)
 Well posedness, condition numbers
 Polynomial based approximations
 Power basis interpolation,
 Lagrange interpolation
 Weierstrass approximation theorem)
 Interpolatory Quadrature rules
 Orthogonal polynomials and Gauss Quadrature Formulas
 L2 projection
 Review of elementary PDEs
 Introduction to Finite Difference Methods
 Introduction to Finite Element Methods
Module 2 (Basis of Numerical Analysis  Part II  Prof. Ganluigi Rozza)
 Least square methods
 Solution methods for Linear Systems
 direct solvers
 iterative solvers
 Eigenvalues/Eigenvectors
 Solution methods for nonLinear systems
 Review of ODEs
Module 3 (Basis of Numerical Modeling  Prof. Gianluigi Rozza)
 Data assimilation in biomechanics, statistics, medicine,  electric signals
 Model order reduction of matrices
 Linear models for hydraulics, networks, logistics
 State equations (real gases), applied mechanics systems,  grow population models, financial problems
 Applications of ODEs
 example in electric phenomena, signals and dynamics of  populations (LotkeVolterra)
 Models for preypredator, population dynamics, automatic  controls
 Applications of PDEs, the poisson problem
 Elastic rope
 Bar under traction
 Heat conductivity
 Maxwell equation
Module 4 (Numerical Analysis with Python  Prof. Luca Heltai)
 Introduction to Python, Numpy, Scipy
 Working with numpy arrays, matrices and ndarrays
 Exercises on Condition numbers, interpolation, quadratures
 Using numpy for polynomial approximation
 Using numpy for numerical integration
 Using numpy/scipy for ODEs
 Solving nonlinear systems of equations
 Using numpy/scipy for simple PDEs
Students projects
Application of the Finite Element Method / Finite Difference Method to the solution of models taken from the course
https://people.sissa.it/~grozza/amnasc/]
Further material provided during lectures by Prof. Gianluigi Rozza [References and Text Books:
 A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathe matics. SpringerVerlag, New York, 2000. [EBookITA] [EBookENG]
 A. Quarteroni. Modellistica Numerica per problemi differenziali. Springer, 2008. [EBookITA]
 A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009. [EBookENG]
 A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer Verlag, 2008. [EBookENG]
 S. Brenner and L. Scott. The mathematical theory of finite element methods. Springer Verlag, 2008. [EBookENG]
 D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk, and M. Fortin. Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006. Springer Verlag, 2008. [EBookENG]
 D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001. [EBookENG]
 A. Quarteroni, F. Saleri, P. Gervasio.* Scientific Computing with Matlab and Octave*. Springer Verlag, 2006. [EBookENG]
 B. Gustaffson* Fundamentals of Scientific Computing, *Springer, 2011 [EBookENG]
 Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. *Elements of Scientific Computing, *Springer, 2010 [EBookENG]
Note that, when connecting from SISSA, all of the text books above are available in full text as pdf files.
MHPC students)
Instructions for git aware students (andThis repository contains, assignements, workspaces, and other material for the course P1.4
New material will be uploaded frequently,
Remember to set a second remote, either to our private seed
git remote add P1.4_seed https://github.com/lucaheltai/numericalanalysis20212022.git
or (if using ssh keys in your github account)
git remote add P1.4_seed git@github.com:lucaheltai/numericalanalysis20202021.git
and to update before the lectures:
git pull P1.4_seed master
Please consider contributing pull requests to correct typos, or better document the material in this repository!
Licencing
The content of this repository is distributed with a Creative Common licence. See the file LICENCE.md in this directory for more information.
Attribution
Some of the material in this repository was adapted from the pythonlectures by Robert Johansson. Take a look at his repository for additional material and lectures.